Browsing by Author "Deniz S."
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Item Implementation of taylor collocation and adomian decomposition method for systems of ordinary differential equations(American Institute of Physics Inc., 2015) Bildik N.; Deniz S.The importance of ordinary differential equation and also systems of these equations in scientific world is a crystal-clear fact. Many problems in chemistry, physics, ecology, biology can be modeled by systems of ordinary differential equations. In solving these systems numerical methods are very important because most realistic systems of these equations do not have analytic solutions in applied sciences In this study, we apply Taylor collocation method and Adomian decomposition method to solve the systems of ordinary differential equations. In these both scheme, the solution takes the form of a convergent power series with easily computable components. So, we will be able to make a comparison between Adomian decomposition and Taylor collocation methods after getting these power series. © 2015 AIP Publishing LLC.Item Comparison of solutions of systems of delay differential equations using Taylor collocation method, Lambert W function and variational iteration method(Sharif University of Technology, 2015) Bildik N.; Deniz S.In this paper, solution of systems of delay differential equations, with initial conditions, using numerical methods, including the Taylor collocation method, the Lambert W function and the variational iteration method, is considered. We have endeavored to show the most appropriate method by comparing the solutions of this system of equations with different types of methods. All numerical computations have been performed on the computer algebraic system, Matlab. © 2015 Sharif University of Technology. All rights reserved.Item Application of adomian decomposition method for singularly perturbed fourth order boundary value problems(American Institute of Physics Inc., 2016) Deniz S.; Bildik N.In this paper, we use Adomian Decomposition Method (ADM) to solve the singularly perturbed fourth order boundary value problem. In order to make the calculation process easier, first the given problem is transformed into a system of two second order ODEs, with suitable boundary conditions. Numerical illustrations are given to prove the effectiveness and applicability of this method in solving these kinds of problems. Obtained results shows that this technique provides a sequence of functions which converges rapidly to the accurate solution of the problems. © 2016 Author(s).Item Applications of optimal perturbation iteration method for solving nonlinear differential equations(American Institute of Physics Inc., 2017) Deniz S.; Bildik N.Perturbation iteration method has been recently constructed and it has been also proven that this technique is very effective for solving some nonlinear differential equations. In this study, we develop the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. This work will greatly improve the computational efficiency of the perturbation iteration method. Applications also show that only a few terms are required to get an approximate solution which is more accurate and efficient than many other methods in literature. © 2017 Author(s).Item Modification of perturbation-iteration method to solve different types of nonlinear differential equations(American Institute of Physics Inc., 2017) Bildik N.; Deniz S.Perturbation iteration method has been recently constructed by Pakdemirli and co-workers. It has been also proven that this technique is very effective and applicable for solving some nonlinear differential equations. In this study we suggest a modification to expedite the solution process of perturbation-iteration algorithms. This work might greatly improve the computational efficiency of the perturbation iteration method and also its Mathematica package to solve nonlinear equations. Numerical illustrations are also given to show how modified method eliminates cumbersome computational work needed by perturbation iteration method. © 2017 Author(s).Item A new efficient method for solving delay differential equations and a comparison with other methods(Springer Verlag, 2017) Bildik N.; Deniz S.In this paper, a new analytical technique, namely the optimal perturbation iteration method, is presented and applied to delay differential equations to find an efficient algorithm for their approximate solutions. Effectiveness of this method is tested by various examples of linear and nonlinear problems of delay differential equations. Obtained results reveal that optimal perturbation iteration algorithm is very effective, easy to use and simple to perform. © 2017, Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg.Item Optimal Perturbation Iteration Method for Solving Nonlinear Heat Transfer Equations(American Society of Mechanical Engineers (ASME), 2017) Deniz S.In this paper, the new optimal perturbation iteration method (OPIM) is introduced and applied for solving nonlinear differential equations arising in heat transfer. The effectiveness of the proposed method will be tested by considering two specific applications: the temperature distribution equation in a thick rectangular fin radiation to free space and cooling of a lumped system with variable specific heat. Comparing different methods shows that the results obtained by optimal perturbation iteration method are very good agreement with the numerical solutions and perform better than the most existing analytic methods. Copyright © 2017 by ASME.Item New analytic approximate solutions to the generalized regularized long wave equations(Korean Mathematical Society, 2018) Bildik N.; Deniz S.In this paper, the new optimal perturbation iteration method has been applied to solve the generalized regularized long wave equation. Comparing the new analytic approximate solutions with the known exact solutions reveals that the proposed technique is extremely accurate and effective in solving nonlinear wave equations. We also show that, un like many other methods in literature, this method converges rapidly to exact solutions at lower order of approximations. © 2018 Korean Mathematial Soiety.Item Optimal perturbation iteration method for Bratu-type problems(Elsevier B.V., 2018) Deniz S.; Bildik N.In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by studying Bratu-type equations. Our results show that only a few terms are required to obtain an approximate solution which is more accurate and efficient than many other methods in the literature. © 2016 The AuthorsItem Comparative Study between Optimal Homotopy Asymptotic Method and Perturbation-Iteration Technique for Different Types of Nonlinear Equations(Springer International Publishing, 2018) Bildik N.; Deniz S.In this paper, we compare optimal homotopy asymptotic method and perturbation-iteration method to solve random nonlinear differential equations. Both of these methods are known to be new and very powerful for solving differential equations. We give some numerical examples to prove these claims. These illustrations are also used to check the convergence of the proposed methods. © 2016, Shiraz University.Item Solving the burgers' and regularized long wave equations using the new perturbation iteration technique(John Wiley and Sons Inc., 2018) Bildik N.; Deniz S.In this study, an efficient framework is provided to handle nonlinear partial differential equations by implementing perturbation iteration method. This method is recovered and amended to solve the Burgers' and regularized long wave equations. Comparing our new solutions with the exact solutions reveals that this technique is extremely accurate and effective in solving nonlinear models. Convergence analysis and error estimate are also supplied using some critical theorems. © 2017 Wiley Periodicals, Inc.Item On a new modified fractional analysis of Nagumo equation(World Scientific Publishing Co. Pte Ltd, 2019) Saad K.M.; Deniz S.; Baleanu D.In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo-Fabrizio and Atangana-Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called h-curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly. © 2019 World Scientific Publishing Company.Item New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques(American Institute of Mathematical Sciences, 2020) Bildik N.; Deniz S.In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations via perturbation iteration technique and newly developed optimal perturbation iteration method. Some specific examples are given and obtained solutions are compared with other methods and analytical results to confirm the good accuracy of the proposed methods.We also discuss the convergence of the optimal perturbation iteration method for partial differential equations. The results reveal that perturbation iteration techniques,unlike many other techniques in literature, converge rapidly to exact solutions of the given problems at lower order of approximations. © 2020 American Institute of Mathematical Sciences. All rights reserved.Item Optimal iterative perturbation technique for solving jeffery–hamel flow with high magnetic field and nanoparticle(Wilmington Scientific Publisher, 2020) Bildik N.; Deniz S.In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeffery–Hamel flow is conducted via the newly developed technique. We use the optimal iterative perturbation method with multiple parameters to see the effects of the magnetic field and nanoparticle on the Jeffery–Hamel flow. Comparing our new approximate solutions with some earlier works proved the excellent accuracy of the newly proposed tech-nique. Convergence analysis of the proposed method is also discussed and error estimation is given to anticipate the accuracy of higher-order approximate solutions. © 2020, Wilmington Scientific Publisher. All rights reserved.Item Optimal perturbation iteration method for solving fractional model of damped burgers' equation(MDPI AG, 2020) Deniz S.; Konuralp A.; la Sen M.D.The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers' equation. The classical damped Burgers' equation is remodeled to fractional differential form via the Atangana-Baleanu fractional derivatives described with the help of the Mittag-Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed. © 2020 by the authors.Item Rational Chebyshev collocation method for solving nonlinear heat transfer equations(Elsevier Ltd, 2020) Deniz S.; Sezer M.In this paper, the classical collocation method has been revisited and modified by using the Chebyshev polynomials for solving nonlinear differential equations. Linear and nonlinear terms are converted to algebraical equations with the aid of the matrix relations. Resulting equations are solved to get unknown coefficients of rational Chebyshev polynomials. We apply the proposed technique for solving nonlinear heat transfer equations. Obtained results reveal that the rational Chebyshev collocation method can be safely applied to different types of nonlinear ordinary differential equations arising in science and engineering. © 2020 Elsevier LtdItem New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods(De Gruyter, 2020) Bildik N.; Deniz S.In this paper, we implement the optimal homotopy asymptotic method to find the approximate solutions of the Poisson-Boltzmann equation. We also use the results of the conjugate gradient method for comparison with those of the optimal homotopy asymptotic method. Our study reveals that the optimal homotopy asymptotic method gives more effective results than conjugate gradient algorithms for the considered problems. © 2020 Walter de Gruyter GmbH, Berlin/Boston.Item Approximate solution of the electrostatic nanocantilever model via optimal perturbation iteration method(John Wiley and Sons Inc, 2021) Adel W.; Deniz S.In this article, a new technique is used to solve the nonlinear boundary value problem of a cantilever-type nanoelectromechanical system. The technique is called the optimal perturbation iteration method and it is used to solve the problem in the form of a nonlinear differential equation with negative power-law nonlinearity. A convergence and error estimation of the proposed method is presented proving that the method is convergent. Results for the application of the proposed technique are demonstrated through two examples and the tables and figures prove that the method is efficient and straightforward. © 2021 John Wiley & Sons Ltd.Item A new efficient technique for solving modified Chua’s circuit model with a new fractional operator(Springer Science and Business Media Deutschland GmbH, 2021) De la Sen M.; Deniz S.; Sözen H.Chua’s circuit is an electronic circuit that exhibits nonlinear dynamics. In this paper, a new model for Chua’s circuit is obtained by transforming the classical model of Chua’s circuit into novel forms of various fractional derivatives. The new obtained system is then named fractional Chua’s circuit model. The modified system is then analyzed by the optimal perturbation iteration method. Illustrations are given to show the applicability of the algorithms, and effective graphics are sketched for comparison purposes of the newly introduced fractional operators. © 2021, The Author(s).Item A new modified semi-analytical technique for a fractional-order Ebola virus disease model(Springer-Verlag Italia s.r.l., 2021) Srivastava H.M.; Deniz S.Ebola virus disease is a fatal hemorrhagic fever of humans and primates caused by viruses. There are many mathematical models to investigate this viral disease. In this paper, the classical form of the Ebola virus disease model has been modified by using new fractional derivatives. The resulting fractional forms of the Ebola virus disease model have then been examined by applying a newly-developed semi-analytical method. The optimal perturbation iteration method has been implemented to obtain new approximate solutions to the system of differential equations which better model the Ebola virus disease. New algorithms are constructed by using three types of operators of fractional derivatives. A real-world problem is also solved in order to prove the efficiency of the proposed algorithms. A good agreement has been found with the real values of the parameters. Finally, several graphical illustrations are presented for different values of the involved biological parameters to show the effects of the new approximate solutions. Obtained results prove that the new method is highly accurate in solving these types of fractional models. © 2021, The Royal Academy of Sciences, Madrid.